Ellipse

Pronunciation: /ɪˈlɪps/ ?

horizontal and vertical ellipses.
Figure 2: Horizontal and vertical ellipses.

A ellipse is a closed curve that can be represented by the equation

(x-h)^2/(a^2)+(y-k)^2/b^2=1,a>b>0
if the major axis is horizontal, or
(y-k)^2/(a^2)+(k-h)^2/b^2=1,a>b>0
if the major axis is vertical.[1]

Something that is related to an ellipse or is in the shape of an ellipse can be called elliptic or elliptical. For example, elliptic geometry is a geometry that can be visualized as taking place on the surface of an ellipse.

Variables and Parameters in Elliptical Equation
Variable or
Parameter
Description
xThe variable x represents the horizontal axis.
yThe variable y represents the vertical axis.
aThe parameter a represents the length of the semimajor axis.
bThe parameter b represents the length of the semiminor axis.
hThe parameter h represents the x position of the center of the ellipse, which is at the intersection of the major axis and minor axis.
kThe parameter k represents the y position of the center of the ellipse, which is at the intersection of the major axis and minor axis.
Table 1

A plane intersecting a cone forming an ellipse.
Figure 1: Ellipse as a conic section. Image courtesy Marcelo Reis. Image licensed under GNU Free Documentation License. Click on the image for more information.

Each ellipse has two axes, a major axis and a minor axis. Each axis is a line around which the ellipse is symmetrical. One is horizontal and the other is vertical. The major axis of an ellipse is the larger of the two axes. The major axis is also called the principal axis. The minor axis of an ellipse is the smaller of the two axes. Half the major axis is called the semimajor axis. Half the minor axis is called the semiminor axis. In the equation for the ellipse, a is the length of the semimajor axis, and b is the length of the semiminor axis.

(h,k) is a point at the center of the ellipse. Changing h moves the ellipse to the left or right. Changing k moves the ellipse up or down.

An ellipse is a conic section formed by intersecting a plane with a cone.

Properties of an Ellipse

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Manipulative 1: Ellipse. Click for classroom demonstrator. Created with GeoGebra.

The endpoints of the horizontal axis are at (h+a,k), (h-a,k). The endpoints of the horizontal ellipse are at (h,k+b), (h,k-b). In manipulative 1, click on the 'Show major axis' to see the major axis and click one the 'show minor axis' to see the minor axis. Click on the blue points on the sliders in manipulative 1 and drag them to change the ellipse.

Each ellipse has two foci. Each focus can be found on the major axis. (h+square root of c,k), (h-square root of c,k) for a horizontal ellipse, or (h,k+square root of c), (h,k-square root of c) where c^2=a^2-b^2. The foci are two points on the major axis of the ellipse. For each point in the ellipse, the sum of the distance of that point to the foci is equal to 2a. Click on the 'Show foci' checkbox in manipulative 1 to show the foci. Click on the blue point on the ellipse and drag it to change the figure.

The eccentricity of an ellipse is a measure of how much it is changed from a circle. In the case where b=a, the ellipse is a circle and the eccentricity is 0. The formula for the eccentricity of an ellipse is

e=square root(1-(b/a)^2)
where b<a. Another form of the equation of the ellipse uses the length of the semimajor axis and the eccentricity:
y^2=(a^2-x^2)(1-e^2)

An ellipse can also be defined with one focus point and a directrix. An ellipse is all points such that the ratio of the distance from the focus to the point on the ellipse divided by the distance from the point on the ellipse to the directrix is equal to the ratio of c to a.

Formulas Related to the Ellipse

NameFormula
Equation of an ellipse if the major axis is horizontal.(x-h)^2/(a^2)+(y-k)^2/b^2=1,a>b>0
Equation of an ellipse if the major axis is horizontal.(y-k)^2/(a^2)+(k-h)^2/b^2=1,a>b>0
Endpoints of the horizontal axis of a horizontal ellipse.(h+a,k), (h-a,k) where c2 = a2 - b2
Endpoints of the vertical axis of a vertical ellipse.(h,k+b), (h,k-b) where c2 = a2 - b2
Foci of a horizontal ellipse.(h+square root of c,k), (h-square root of c,k) where c2 = a2 - b2
Foci of a vertical ellipse.(h,k+square root of c), (h,k-square root of c) where c2 = a2 - b2
Eccentricity of an ellipse.e=square root(1-(b/a)^2)
Area of an ellipse.pi*a*b where a and b are the lengths of the major and minor axes of the ellipse.
The circumference of an ellipse is expressed using an infinite series.C=2*pi*a[1-(1/2)^2*e^2-((1*3)/(2*4))^2*((e^4)/3))-((1*3*5)/(2*4*6))^2*((e^6)/5)-...] where ε is the eccentricity.
Table 2 - Formulas related to an ellipse.

Sketching an Ellipse

The general form of the elliptical equation is

(x-h)^2/a^2+(y-k)^2/a^2=1.
The sketch will be of the elliptical equation
(x-1)^2/9+(y+2)^2/4=1.
StepExampleDescription
1Cartesian coordinate system with point (1,-2) plotted First, plot the center of the ellipse at (h,k). In this example (h,k)=(1,-2).

For the x-coordinate, start with x-h=x-1. Now subtract x from both sides to get -h=-1. Multiply both sides by -1 to get h=1.

For the y-coordinate, start with y-k=y+2. Now subtract y from both sides to get -k=2. Multiply both sides by -1 to get k=-2.

2Cartesian coordinate system with points (-2,-2) and (4,-2) plotted Now plot the endpoints of the horizontal axis. The left endpoint is at (h-a,k). In this case that is (1-3,-2)=(-2,-2). Plot the right endpoint at (h+a,k). For this equation, (1+3,-2)=(4,-2).
3Cartesian coordinate system with points (1,1) and (1,-5) plotted Now plot the endpoints of the horizontal axis. The bottom endpoint is at (h,k-b). In this case that is (1,-2-2)=(1,-4). Plot the top endpoint at (h,k+b). For this equation, (1,-2+2)=(1,0).
4Cartesian coordinate system with ellipse (x-1)^2/9+(y+2)^2/9=1 plotted Now sketch the ellipse.
5Cartesian coordinate system with ellipse (x-1)^2/9+(y+2)^2/9=1 plotted with foci To plot the foci, calculate the value of c. The equation for c is c^2=a^2-b^2. Using the values of a and b, c^2=9-4=5, and c=square root of 5 is about 2.24.. Since the major axis is the horizontal axis, the foci are at (h+-c,k). Substituting the values gives (1-2.24,-2)=(-1.24,-2) and (1+2.24,-2)=(3.24,-2).
Table 2

References

  1. ellipse. http://wordnet.princeton.edu/. WordNet. Princeton University. (Accessed: 2011-01-08). http://wordnetweb.princeton.edu/perl/webwn?s=ellipse&sub=Search+WordNet&o2=&o0=1&o7=&o5=&o1=1&o6=&o4=&o3=&h=.
  2. Author unknown. The Elements of the Conic Sections with the Sections of the Conoids, 3rd edition, pp 19-40. Published by J. Deighton and Sons, 1826. (Accessed: 2010-01-07). http://www.archive.org/stream/elementsofconics00hastuoft#page/19/mode/1up/search/ellipse.

Printed Resources

Cite this article as:


Ellipse. 2010-01-07. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/e/ellipse.html.

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Revision History


2010-01-07: Added "References" (McAdams, David.)
2009-11-12: Added table of formulas. (McAdams, David.)
2009-03-10: Added definition of axis of an ellipse (McAdams, David.)
2008-12-10: Initial version (McAdams, David.)

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